Generated by Claude (Sonnet 4.5) on 2025-10-13
We want to find whether commutative hyperoperators can be extended to fractional ranks (e.g., operation 2.5, halfway between multiplication and commutative exponentiation) that would:
Schröder's Equation provides a framework for fractional iteration: - For function f with fixed point and multiplier s - Solve: Ψ(f(x)) = s·Ψ(x) - Then fractional iterates: f^(t)(x) = Ψ(-1)(st · Ψ(x))
Abel's Equation: α(f(x)) = α(x) + 1 - Related to Schröder via: Ψ(x) = s^(α(x)) - Also gives fractional iteration
The commutative hyperoperators are defined:
⊕₀(x, y) = log(exp(x) + exp(y))
⊕₁(x, y) = x + y
⊕₂(x, y) = x × y
⊕₃(x, y) = exp(log(x) × log(y))
⊕ₙ(x, y) = exp(⊕ₙ₋₁(log(x), log(y))) [for n > 3]
Key property we proved: ln(⊕ₙ(x, y)) = ⊕ₙ₋₁(ln(x), ln(y))
Open question from Ghalimi: Can we define ⊕ₙ for n ∈ ℝ or ℂ?
Define ⊕_{n+t} for 0 < t < 1 using a smooth interpolation that satisfies:
Naive attempt: Linear interpolation in the transform space?
⊕_{n+t}(x, y) = exp((1-t)·⊕_{n-1}(log(x), log(y)) + t·⊕ₙ(log(x), log(y)))
Problem: This probably doesn't preserve the recursive structure or other properties.
For a fixed y, consider the function:
fᵧ(x) = ⊕ₙ(x, y)
Try to solve Schröder's equation for this function, then use that to define fractional operations.
Challenge: The multiplier and fixed point structure may not be straightforward.
The recursive structure suggests:
⊕ₙ₊₁ = exp ∘ ⊕ₙ ∘ (log × log)
Can we define "fractional composition" of the exp and log operations?
If we could define exp^(t) for 0 < t < 1 (fractional iteration of exp), then:
⊕_{n+t} = exp^(t) ∘ ⊕ₙ ∘ (log × log)^(t)
But: Fractional iteration of exp is notoriously difficult because exp has no fixed point in ℝ.
For small arguments, expand ⊕ₙ(x, y) as a power series in log(x) and log(y), then interpolate the coefficients smoothly as functions of n.
Challenge: Ensuring convergence and preserving properties.
At each level n ≥ 3, we want:
⊕ₙ(x, y)
/ \
/ \
ln(·) ←→ ⊕ₙ^(1/k)(x)
Where ⊕ₙ^(1/k) is a kth "root" operation satisfying:
⊕ₙ(⊕ₙ^(1/k)(x), ..., ⊕ₙ^(1/k)(x)) = x [k times]
Critical property needed:
ln(⊕ₙ^(1/k)(x)) = (1/k) · something related to ln(x)
This would allow the mean relationship:
exp(generalized_mean_{n-1}(ln(odds))) = generalized_mean_n(odds)
We want ⊕{2.5} such that: - ⊕{2.5}(⊕{2.5}(x, y), ⊕{2.5}(x, y)) ≈ ⊕₃(x, y) - ⊕{2.5}(x, y) is between x×y and exp(log(x)×log(y)) - ln(⊕{2.5}(x, y)) = ⊕_{1.5}(ln(x), ln(y))
Computational test: For x=2, y=3: - ⊕₂(2, 3) = 6 - ⊕₃(2, 3) = exp(log(2)×log(3)) ≈ 1.933 - ⊕_{2.5}(2, 3) should be ≈ 3.29 (geometric mean of 6 and 1.933)
For ⊕₃ (commutative exponentiation), what is ⊕₃^(1/2)?
We want: ⊕₃(⊕₃^(1/2)(x), ⊕₃^(1/2)(x)) = x
So: exp(log(y)×log(y)) = x where y = ⊕₃^(1/2)(x) Therefore: y = exp(√(log(x)/log(y)))
This is an implicit equation! We'd need to solve for y.
Existence: Do fractional commutative hyperoperators exist that satisfy all desired properties?
Uniqueness: If they exist, are they unique? (Fractional iteration often has non-unique solutions)
Analyticity: Can they be extended to a holomorphic function of n?
Computation: Can they be computed efficiently? (Might require numerical solution of functional equations)
Triangle structure: Do fractional operations automatically give us the "third vertex" of the triangle at each level?
Mean relationships: If we can define fractional operations, do we automatically get generalized mean relationships?
Conjecture: There exists a family of operations ⊕ₙ for n ∈ ℝ, n ≥ 0, such that:
If true, this would enable a "tower of triangles" with mean relationships at every level.
But: Proving existence and constructing it explicitly are major open problems.
This investigation was sparked by observing: - The "triangle of power" (exp, log, √) at the exponentiation level - Distributivity: log(a×b) = log(a) + log(b) - These together enable: exp(arithmean(log(odds))) = geommean(odds)
For this pattern to extend to all levels with commutative hyperoperators, we need both: 1. ✓ Distributivity at all levels (we proved this) 2. ? Fractional operations to complete the triangles (still open)
This appears to be novel mathematics if properly worked out, though the ingredients exist in the literature. The specific question of whether fractional commutative hyperoperators complete the "triangle of power" at each level seems unexplored.
This deserves a proper mathematical investigation, possibly: 1. A computational exploration (try to construct ⊕_{2.5} numerically) 2. An analytical approach using functional equations 3. Reaching out to researchers in fractional iteration / tetration
If this works out, it would be a beautiful unification of: - Generalized means (Hölder) - Commutative hyperoperators (Ghalimi) - Fractional iteration (Schröder/Abel) - The arithmetic-geometric mean connection
Added 2025-10-13 after further discussion
We realized there was confusion between two different notions of "fractional":
Finding operations ⊕{n+t} for non-integer n+t (like ⊕{2.5}, halfway between multiplication and commutative exponentiation).
Status: Theoretically possible via Schröder/Abel methods, no explicit formulas known for Ghalimi's construction.
For each integer level n, finding an operation ⊕ₙ^(1/k) that acts like an "nth root" does for exponentiation.
Key requirement: ln(⊕ₙ^(1/k)(x)) should relate nicely to ln(x), allowing mean calculations.
Status: Unknown whether this is even possible.
At the exponentiation level (n=2), we have a triangle of power:
exp(x)
/ \
/ \
log(x) ←→ x^(1/n)
With the crucial property: log(x^(1/n)) = (1/n)·log(x)
This property enables: exp(arithmetic_mean(log(odds))) = geometric_mean(odds)
For higher levels, we'd need analogous triangles:
⊕ₙ(x, y)
/ \
/ \
ln(·) ←→ ⊕ₙ^(1/k)(·)
Path A: Via Fractional Ranks - Define ⊕_{n+t} for all real n - Hope that this automatically gives us root operations ⊕ₙ^(1/k) - Challenge: No explicit construction known
Path B: Define Roots Directly - At each integer level n, separately define what ⊕ₙ^(1/k) means - Don't require fractional ranks between levels - Challenge: Unclear if this is possible with current structure
For n ≥ 4, commutative hyperoperators enter the complex domain: - ln(ln(2)) < 0, so we need complex numbers - Complex logarithms are multi-valued (branch cuts) - Complex roots are multi-valued (n different nth roots) - The triangle relationships might only hold for specific branch choices
Possible simplification: Restrict to positive reals x, y > e?
For x, y > e: - ⊕₃(x, y) stays real and positive - ⊕₄(x, y) = exp(⊕₃(ln(x), ln(y))) might stay real - Need to verify this computationally
Proven: 1. Distributivity holds at all levels: ln(⊕ₙ(x, y)) = ⊕ₙ₋₁(ln(x), ln(y)) 2. This works even in ℂ (by construction)
Unknown: 1. Whether fractional ranks exist with nice properties 2. Whether root operations ⊕ₙ^(1/k) can be defined 3. Whether these would complete the triangle structure 4. Whether this enables mean relationships at higher levels
The arithmetic-geometric mean relationship works because of two independent structures: 1. Distributivity: log(a×b) = log(a) + log(b) 2. Triangle: log, exp, and roots interrelate via log(x^(1/n)) = (1/n)·log(x)
We have (1) at all levels for commutative hyperoperators.
Open question: Can we construct (2) at all levels?
This might require: - Novel mathematics - Numerical experimentation - Better understanding of fractional iteration - Or it might simply be impossible
Likelihood of success: Uncertain but worth exploring
Potential impact: High - would unify several mathematical structures
Difficulty: Significant - requires solving open problems in fractional iteration
Time investment: Weeks to months of serious mathematical work
This is a promising research direction but not a quick win.